System Bits: March 7

Math picture languageHarvard University researchers reminded that Galileo called mathematics the “language with which God wrote the universe,” as he described a picture-language. Now that language has a new dimension.

Arthur Jaffe (left) and Zhengwei Liu are the creators of a new, 3D pictorial language for mathematics. They believe this new language could lead to important new insights in math, physics and a host of other subjects.(Source: Harvard University)

In a development that holds the potential as a tool across a range of topics, from pure math to physics to quantum information, Arthur Jaffe, the Landon T. Clay Professor of Mathematics and Theoretical Science, postdoctoral fellow Zhengwei Liu, and researcher Alex Wozniakowski — all of Harvard University – have developed a 3D picture-language for mathematics.

Called quon, the researchers believe the language holds promise for being able to transmit not only complex concepts, but also vast amounts of detail in relatively simple images.

Jaffe said the paper is the result of work that the team has been doing for the past year and a half, and they regard it as the start of something new and exciting. “It seems to be the tip of an iceberg. We invented our language to solve a problem in quantum information, but we have already found that this language led us to the discovery of new mathematical results in other areas of mathematics. We expect that it will also have interesting applications in physics.”

When it comes to the “language” of mathematics, humans start with the basics — by learning their numbers. As we get older, however, things become more complex. Liu said we learn to use algebra, and we use letters to represent variables or other values that might be altered. “Now, when we look at research work, we see fewer numbers and more letters and formulas. One of our aims is to replace ‘symbol proof’ by ‘picture proof.’” The new language relies on images to convey the same information that is found in traditional algebraic equations — and in some cases, even more.

This pictorial language for mathematics can give insights and a way of thinking that isn’t seen in the usual, algebraic way of approaching mathematics, Jaffe said in an article on the Harvard website. “For centuries there has been a great deal of interaction between mathematics and physics because people were thinking about the same things, but from different points of view. When we put the two subjects together, we found many new insights, and this new language can take that into another dimension.”

Among their pictorial feats, Jaffe said, are the complex equations used to describe quantum teleportation. The researchers have pictures for the Pauli matrices, which are fundamental components of quantum information protocols. This shows that the standard protocols are topological, and also leads to discovery of new protocols.
“It turns out one picture is worth 1,000 symbols,” Jaffe said.
“We could describe this algebraically, and it might require an entire page of equations,” Liu added. “But we can do that in one picture, so it can capture a lot of information.”

Uninitialized-use ‘bugs’ may provide hacker portal into Linux
According to new research from the Georgia Institute of Technology, although popular with programmers the world over for its stability, flexibility and security, Linux now appears to be vulnerable to hackers.

The team explained that uninitialized variables ­– largely overlooked bugs mostly regarded as insignificant memory errors – are actually a critical attack vector that can be reliably exploited by hackers to launch privilege escalation attacks in the Linux kernel, and when successful, these intrusions give attackers increasing levels of access to a network’s resources.

The lead researcher on the project, Georgia Tech Ph.D. student Kangjie Lu, said while other kernel bugs and vulnerabilities have been examined and remedied, uninitialized-use bugs are not well studied, and to date, no practical defense mechanisms have been developed to protect against these attacks. In fact, despite potentially dangerous consequences, uninitialized-use bugs are seldom even classified as security vulnerabilities.

To prove that these bugs do present a security risk, researchers developed a novel approach, known as targeted stack spraying, to attack the operating system (OS) kernel. Along with a technique that occupies large portions of the memory to control the stack, the automated attack probes the stack to find weaknesses that user-mode programs can exploit to direct kernel code paths and leave attacker-controlled data on the kernel stack. Ultimately, the goal of this attack is to reliably control the value of a specific uninitialized variable in the kernel space of a running program, the team explained.

The research findings confirm that hackers using this method can automatically prepare a malicious pointer in the uninitialized variable. When the malicious pointer is used, a privilege escalation attack targeting the Linux kernel may occur.

Interestingly, the research showed that utilizing the targeted stack-spraying approach allows attackers to reliably control more than 91 percent of the Linux kernel stack, which, in combination with uninitialized-use vulnerabilities, suffices for a privilege escalation attack.

To combat this, the researchers also developed a potential solution to the problem that leverages the fact that uninitialized-use attacks usually control an uninitialized pointer to achieve arbitrary read/write/execution.

Accelerating electrons
Under certain specialized conditions, electrons can speed through a narrow opening in a piece of metal more easily than traditional theory says is possible, according to new research by physicists at MIT and the Weizmann Institute in Rehovot, Israel.

New research shows that electrons passing through a narrow constriction in a piece of metal can move much faster than expected, and that they move faster if there are more of them — a seemingly paradoxical result. In this illustration, the orange surface represents the potential energy needed to get an electron moving, and the “valley” at center represents the constricted portion. (Source: MIT)

MIT physics professor Leonid Levitov, who is the senior author of a paper describing the finding that appears this week in the Proceedings of the National Academy of Sciences, said this “superballistic” flow resembles the behavior of gases flowing through a constricted opening, however it takes place in a quantum-mechanical electron fluid. In these constricted passageways, whether for gases passing through a tube or electrons moving through a section of metal that narrows to a point, it turns out that the more, the merrier: Big bunches of gas molecules, or big bunches of electrons, move faster than smaller numbers passing through the same bottleneck.

The result is that, through a sufficiently narrow, point-like constriction in a metal, electrons can flow at a rate that exceeds what had been considered a fundamental limit, known as Landauer’s ballistic limit. Because of this, the team has dubbed the new effect “superballistic” flow. This represents a great drop in the electrical resistance of the metal — though it is much less of a drop than what would be required to produce the zero resistance in superconducting metals. However, unlike superconductivity, which requires extremely low temperatures, the new phenomenon may take place even at room temperature and thus may be far easier to implement for applications in electronic devices, the researchers said.